Mathematical Sciences: RUI: Spectral Problems for Toeplitz and Other Structured Matrices

当n较大时，如果指定H中的任意n × n成员所需的独立参数的数量与n近似成正比，则可以有效地构造H族方阵。Toeplitz， Hankel和Toeplitz- + Hankel矩阵是有效结构矩阵的例子，正如T. Kailath和他的合著者所定义的给定位移秩的矩阵一样。提出的研究的目的是(a)开发快速算法，用于查找一般类别的有效结构化厄米矩阵的单个特征值-特征向量对，类似于首席研究员的Toeplitz和Toeplitz- + hankel矩阵的算法；(b)研究Toeplitz矩阵的谱性质（特别是实对称Toeplitz矩阵的奇偶谱关系）；(c)研究设计快速算法求非对称Toeplitz矩阵实特征值的可能性；(d)考虑Toeplitz-Sturm矩阵的谱性质。高效结构的厄米矩阵出现在许多重要的物理和科学应用中，包括涉及基于对随机变量过去值的观察来预测其未来值的统计问题，信号处理（从传输信号中提取信息的科学），和地球物理应用，包括地球的自由振荡和地震现象，如地震的研究。在研究这些问题时，常常需要确定非常高阶（千阶）高效结构矩阵的个别特征值。所提出的研究的主要目标之一是开发数值方法来找到这些特征值的方法，这些方法的复杂性（计算要求）与矩阵的二阶成正比，而不是与阶的三次方成正比，因为它是标准计算方法，不利用矩阵的特殊结构。

A family H of square matrices is efficiently structured if the number of independent parameters required to specify an arbitrary n by n member of H is approximately proportional to n when n is large. Toeplitz, Hankel, and Toeplitz-plus-Hankel matrices are examples of efficiently structured matrices, as are matrices of a given displacement rank, as defined by T. Kailath and his coauthors. The purposes of the proposed research are (a) to develop fast algorithms for finding individual eigenvalue-eigenvector pairs for general classes of efficiently structured Hermitian matrices, analogous to the principal investigator's algorithms for Toeplitz and Toeplitz-plus-Hankel matrices; (b) to study spectral properties of Toeplitz matrices (especially, the relationship between the even and odd spectra of real symmetric Toeplitz matrices); (c) to investigate the possibility of devising fast algorithms for finding the real eigenvalues of nonsymmetric Toeplitz matrices; and (d) to consider the spectral properties of Toeplitz-Sturm matrices. Efficiently structured Hermitian matrices arise in many important physical and scientific applications, including statistical problems involving prediction of future values of a random variable based on observations of its past values, signal processing (the science of extracting information from transmitted signals), and geophysical applications including the study of free oscillations of the Earth and seismic phenomena such as earthquakes. In studying these problems it is often necessary to determine individual eigenvalues of efficiently structured matrices of very high order (in the thousands). One of the main objectives of the proposed research is to develop numerical methods for finding these eigenvalues by methods whose complexity (computational requirements) is proportional to the second power of the order of the matrix, rather than to the third power of the order, as it is for standard computational methods that do not take advantage of the special structure of the matrix.